# Simon Kirwan Donaldson

### Quick Info

Cambridge, England

**Simon Donaldson**is an English mathematician who received a Fields Medal for his work on 4 dimensional manifolds. He was knighted in 2012 and awarded the most prestigious mathematical prizes including the Shaw Prize, the Breakthrough Prize in Mathematics, the Oswald Veblen Prize, and the Wolf Prize in Mathematics.

### Biography

**Simon Donaldson**'s father was an engineer whose first career was in the navy. At about the time he married, he changed jobs to work in the Physiology Laboratory in the University of Cambridge. There he constructed apparatus for experiments which were aimed at understanding the nervous system. Simon's mother was brought up in Cambridge and graduated in Natural Sciences from the University of Cambridge but after marrying never had a career. Simon was one of his parents' four children, having an older brother, an older sister and a younger brother. Both his brothers became engineers.

The home in which Simon grew up was one in which people had projects. His father built model airplanes, saw about repairing things in the home and generally was busy working on some project. Simon said [8]:-

I have an early memory of him saying with relish: "... and then I shall be able to get back to research" (presumably, after completing some chores which he had described to me). I had no idea what "research" might be, but from that time the word was tinged with glamour and romance.It was a home in which everyone was busy and active. His maternal grandfather also had a big influence on the children's education. He was a retired schoolmaster and took a great interest in the education of all four children, but perhaps Simon got a little more of his attention than the other three children.

Simon attended a Preparatory School in Cambridge where his favourite subject was history. He saw his future education as progressing to an independent Cambridge Secondary School but this was not to be since when he was twelve year old the family moved to a new home near Sevenoaks, around 30 km south of London. The move came about because his father left his job at the University of Cambridge, going with some others who he had worked with there to a new research unit in London funded by the Medical Research Council. He was part of a team developing neurological implants. The move meant a change of school for Simon and so his secondary school education was at Sevenoaks School in Sevenoaks, Kent which he attended from 1970 to 1975.

The Sevenoaks School was independent but, because there was no state grammar school in Sevenoaks, about half the pupils were funded by the state and half were private. This meant it had a much greater social mix than a normal independent school. At first Simon was not very happy at the school and it took him a few years to settle in. One thing that helped was his passion he had for designing yachts. He did not have a love of practical projects like the rest of his family, so his interest in yacht design was theoretical. This interest helped him fit into the school, since it had a sailing tradition. He read books to help him understand the theory behind the design of ships and these contained more mathematics than he had learnt at school. His father taught him some mathematical techniques and he began to study calculus well before it was taught at school.

By the age of fourteen he had given up designing yachts and was studying mathematics on his own from books that his maternal grandfather bought for him. He wanted to be a mathematician but had a backup plan to become an accountant. While at Sevenoaks School he won a scholarship to study mathematics and physics at Pembroke College, Cambridge which he entered in 1976. There he studied analysis, topology and mathematical physics but the topic he liked most was geometry, although there was not much of that topic in the Cambridge syllabus. He was awarded his B.A. in 1979 and continued to study Part III of the tripos at Cambridge until 1980. One of his tutors at Cambridge described him as a very good student but certainly not the top student in his year. Frank Adams, however, was very impressed with the work Donaldson did in his final examinations and wrote to him to congratulate him on his solutions. Frank Adams was one of his referees for his application to study for a doctorate at Oxford.

In 1980 Donaldson began postgraduate work at Worcester College, Oxford, first under Nigel Hitchin's supervision. It was, however, the activity and culture that developed around Michael Atiyah and Roger Penrose that he found most attractive. Hitchin suggested that he look at a conjecture that he and Atiyah had proposed a couple of years earlier. Donaldson attacked the problem using both methods from partial differential equations and from topology. After a year Hitchin suggested that he changed to become Atiyah's student. He soon made a remarkable breakthrough. Atiyah writes in [3]:-

In 1982, when he was a second-year graduate student, Simon Donaldson proved a result that stunned the mathematical world.This result was published by Donaldson in a paper

*Self-dual connections and the topology of smooth 4-manifolds*which appeared in the

*Bulletin*of the American Mathematical Society in 1983. Atiyah continues his description of Donaldson's work [3]:-

Together with the important work of Michael Freedman ..., Donaldson's result implied that there are "exotic" 4-spaces, i.e. 4-dimensional differentiable manifolds which are topologically but not differentiably equivalent to the standard Euclidean 4-space $\mathbb{R}^{4}$. What makes this result so surprising is that n = 4 is the only value for which such exotic n-spaces exist. These exotic 4-spaces have the remarkable property that (unlike $\mathbb{R}^{4}$) they contain compact sets which cannot be contained inside any differentiably embedded 3-sphere!After being awarded his doctorate from Oxford in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford. He spent the academic year 1983-84 at the Institute for Advanced Study at Princeton. During this year he made a trip to the University of Maryland where he met Nora. She had been awarded a Bachelor of Arts degree from the Universidad del Valle in Colombia in 1975, followed by a Master in Science degree from Virginia Tech in 1981. When Donaldson met her at the University of Maryland she was studying for a Ph.D. in Mathematical Statistics. They married and have three children Andres, Jane and Nicholas. He also has a step-daughter Adriana. Nora was awarded a Ph.D. by the University of Maryland in 1988 for the thesis

*A non-parametric estimation of the tumour onset time in a serial screening experiment*.

After returning to Oxford he continued to hold his Post-Doctoral Research Fellowship at All Souls College until 1985. He became a Quondam Fellow of All Souls College in 1985 and, in that year, was appointed Wallis Professor of Mathematics at St Anne's College Oxford. He spent the year 1997-8 in Stanford before moving to Imperial College, London in 1999. His wife Nora was appointed head of the Biostatistics Unit in the Research and Development Department at King's College Hospital in 1999. He continues to work at Imperial College as Royal Society Research Professor, but since 2014 he is also a permanent member of the Simons Center for Geometry and Physics at Stony Brook University in the United States. Nora Donaldson worked at King's College Hospital until 2005 when she joined the Dental Institute as a Reader in Biostatistics.

Donaldson has received many honours for his work. He received the Junior Whitehead Prize from the London Mathematical Society in 1985. In the following year he was elected a Fellow of the Royal Society and, also in 1986, he received a Fields Medal at the International Congress of Mathematicians at Berkeley. He was presented with the medal by Lars Ahlfors. Three years earlier he had delivered the talk

*Gauge theory and topology*at the Congress in Warsaw, and at the Berkeley Congress in 1986 he gave the plenary address

*The geometry of 4-manifolds*. We note also that in 1998 he was the Vice-President of the International Mathematical Union and delivered the lecture

*Lefschetz Fibrations in Symplectic Geometry*to the 1998 International Congress of Mathematicians in Berlin. He was also a plenary lecturer at the International Congress of Mathematical Physics in 1990 and at the European Congress of Mathematicians in 1992.

In 1991 Donaldson received the William Hopkins Prize from the Cambridge Philosophical Society and, in the following year, the Royal Medal from the Royal Society. He was awarded the London Mathematical Society's Polya prize in 1999 and some years later, the 2006 King Faisal Prize. He also received the Crafoord Prize from the Royal Swedish Academy of Sciences in 1994 [5]:-

... for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants ...Atiyah describes the contribution which led to Donaldson's award of a Fields Medal in [3]. He sums up Donaldson's contribution:-

When Donaldson produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration. Slowly the message has gotten across and now Donaldson's ideas are beginning to be used by others in a variety of ways. ... Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry.The article [7] is very interesting and provides both a collection of reminiscences by Donaldson on how he came to make his major discoveries while a graduate student at Oxford and also a survey of areas which he has worked on in recent years. Donaldson writes in [7] that nearly all his work has come under the headings:-

(1) Differential geometry of holomorphic vector bundles.

(2) Applications of gauge theory to 4-manifold topology.

and he relates his contribution to that of many others in the field.
(2) Applications of gauge theory to 4-manifold topology.

Donaldson's work in summed up by R Stern in [36]:-

In 1982 Simon Donaldson began a rich geometrical journey that is leading us to an exciting conclusion to this century. He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds.Donaldson was elected to the National Academy of Sciences in 2000. In February 2006 he was awarded the King Faisal International Prize for science for [43]:-

... seminal contributions to theories which have strengthened the links between mathematics and physics, and helped provide a rigorous foundation for physical theories giving a very good description of the laws of matter at the sub-nuclear level.In April 2008, he was awarded the Frederic Esser Nemmers Prize in Mathematics from Northwestern University. The Prize was given for his [44]:-

... ground-breaking work in four-dimensional topology, symplectic geometry and gauge theory, and for his remarkable use of ideas from physics to advance pure mathematics.John Franks, the Chair of Mathematics at Northwestern University, gave more details of Donaldson's contributions which led to the award [44]:-

Donaldson's breakthrough work developed new techniques in the geometry of four-manifolds and the study of their smooth structures. His methods have been described as extremely subtle, using difficult nonlinear partial differential equations. Using instantons, solutions to the equations of Yang-Mills gauge theory, he gained important insight into the structure of closed four-manifolds. Gauge theory techniques also enabled him to show the existence of four-manifolds with no smooth structure and others with infinitely many. His work has provided the seminal steps for the work of others in study of four-manifolds.In 2009 Donaldson, together with Clifford H Taubes, was awarded the Shaw Prize in Mathematical Sciences. The Committee who made the award wrote that Donaldson and Taubes [45]:-

... are the two geometers who have transformed the whole subject by pioneering techniques and ideas originating in theoretical physics, including quantum theory [and] have totally changed our geometrical understanding of space and time.For more information about this award, including further extracts from the citation for Donaldson, see THIS LINK.

Donaldson was knighted in 2012, the same year that he was elected as a fellow of the American Mathematical Society. His next major award was the Breakthrough Prize in Mathematics which he received in 2015 [26]:-

... for the new revolutionary invariants of four-dimensional manifolds and for the study of the relation between stability in algebraic geometry and in global differential geometry, both for bundles and for Fano varieties.The Breakthrough Prize was launched in 2012 to honour important, primarily recent, achievements in Fundamental Physics (first awards 2012), Life Sciences (first awards 2013) and Mathematics (first awards 2015):-

"All is number," taught Pythagoras. Though modern mathematics encompasses far more than numbers alone, the principle remains true. Mathematics is the universal language of nature. Mathematics is also fundamental to the growth of knowledge, as it is the scaffolding that supports all the sciences. Its relationship to physics is particularly intimate. From imaginary numbers to Hilbert spaces, what once seemed pure abstractions have turned out to underlie real physical processes. In addition, all fields in the life sciences today utilise the power of statistical and computational approaches to research. The mathematics prizes reward significant discoveries across the many branches of the subject. They were founded by Yuri Milner and are funded by grants from the foundations established by Yuri and Julia Milner.The 2015 Breakthrough Prize ceremony was held on 9 November 2014 at NASA's Hangar 1 in Mountain View, California; this is the home of the Kepler Mission to find planets capable of supporting life. To launch the mathematics award, five winners were named each receiving $3 million. All five recipients of the Prize agreed to serve on the Selection Committee, responsible for choosing subsequent winners of the prize from the pool of contenders nominated by the mathematics community. The 2015 ceremony, with many actors and other celebrities present, was broadcast on several television channels.

For the full citation and Donaldson's response on receiving the award, see THIS LINK.

In 2019 Donaldson was one of three collaborators who won the Oswald Veblen Prize. The 2019 Oswald Veblen Prize in Geometry was [1]:-

... awarded to Xiuxiong Chen, Simon Donaldson - both of Stony Brook University - and Song Sun, University of California, Berkeley, for their three-part series, "Kähler-Einstein metrics on Fano manifolds, I, II and III," published in 2015 in the Journal of the American Mathematical Society, in which they proved a long-standing conjecture in differential geometry.Donaldson was a joint winner of the Wolf Prize in Mathematics in 2020 along with Yakov Eliashberg [28]:-

... for their contributions to differential geometry and topology.The citation for Donaldson ends [28]:-

Professor Simon Donaldson is awarded the Wolf Prize for his leadership in geometry in the last 35 years. His work has been a unique combination of novel ideas in global non-linear analysis, topology, algebraic geometry, and theoretical physics, following his fundamental work on 4-manifolds and gauge theory. Especially remarkable is his recent work on symplectic and Kähler geometry.For the full citation, see THIS LINK.

Let us end this biography by quoting Donaldson's own description of his contributions from his National Academy of Sciences web page [30]:-

My research interests lie in the area of mathematics bordering geometry, topology, and analysis and having substantial connections with mathematical physics. Much of my early work hinged on the application of the instanton solutions of the Yang-Mills equations - first introduced in particle physics - as tools to solve purely mathematical problems about the topology of four-dimensional manifolds. This has led to novel and wide-ranging results, not obtainable by other methods, that give a glimpse of the special nature of four-dimensional topology and geometry. More recently my work in this direction has focused on the special class of symplectic manifolds. I have shown that certain classical techniques from complex algebraic geometry can be adapted to this setting and am currently pursuing the implications of this for the classification of symplectic manifolds. Another theme running through my research is the study of certain partial differential equations arising in complex differential geometry. In the 1980s I worked on equations related to holomorphic vector bundles and have recently been studying the application of similar ideas to Kähler metrics.

### References (show)

- 2019 Oswald Veblen Prize in Geometry to Xiuxiong Chen, Simon Donaldson, and Song Sun, News,
*American Mathematical Society*(19 November 2018).

http://www.ams.org/news?news_id=4705 - Biography in
*Encyclopaedia Britannica*. http://www.britannica.com/biography/Simon-Kirwan-Donaldson - M Atiyah, On the work of Simon Donaldson,
*Proceedings of the International Congress of Mathematicians, Berkeley, 1986***1**(Providence, RI, 1987), 3-6. - Autobiography of Simon K Donaldson,
*The Shaw Foundation*(7 October 2009).

https://www.shawprize.org/autobiography/simon-k-donaldson/ - Citation, Crafoord Prize - Mathematics and Astronomy,
*Royal Swedish Academy of Sciences*(1994).

https://www.kva.se/en/prize-laureate/simon-donaldson-2/ - Crafoord Prize to one of the world's foremost mathematicians,
*Royal Swedish Academy of Sciences*(25 January 2001).

https://www.kva.se/en/news/crafoordpriset-till-en-av-varldens-framsta-matematiker-2/ - S Donaldson, Remarks on gauge theory, complex geometry and 4-manifold topology, in
*M Atiyah and D Iagolnitzer (eds.), Fields Medallists Lectures*(Singapore, 1997), 384-403. - S K Donaldson, Simon Donaldson, in
*M Cook, Mathematicians. An outer view of the inner world*(Princeton University Press, 2009), 66-67. - Donaldson, Sir Simon Kirwan,
*Who's Who 2023*(Oxford University Press, 2023). - Donaldson, Simon K, ICM Plenary and Invited Speakers,
*International Mathematical Union*.

https://www.mathunion.org/icm-plenary-and-invited-speakers?combine=Donaldson - M Furuta, The contributions of Simon Kirwan Donaldson (Japanese),
*Sugaku***39**(1) (1987), 16-25. - N Hitchin, Review: The geometry of four-manifolds, by S K Donaldson and P B Kronheimer,
*Bulletin of the American Mathematical Society (N.S.)***28**(2) (1993), 415-418. - I Kra, Review: Riemann surfaces, by S K Donaldson,
*Bulletin of the American Mathematical Society (N.S.)***49**(3) (2012), 455-463. - J Mi, The work of mathematicians awarded the Fields Medal in 1983 and 1986 (Chinese),
*J. Northwest Univ.***19**(1) (1989), 103-104. - V Munoz, Review: Floer homology groups in Yang-Mills theory, by Simon K Donaldson,
*Mathematical Reviews*MR1883043**(2002k:57078)**. - New foreign members elected to the academy,
*Royal Swedish Academy of Sciences*(26 May 2010). - M Pachter, The HLF Portraits: Simon Donaldson,
*Heidelberg Laureate Forum Foundation*(2020).

https://av.tib.eu/media/46784 - Professor Sir Simon Donaldson FRS,
*Imperial College London*.

https://www.imperial.ac.uk/people/s.donaldson - Professor Sir Simon Donaldson,
*All Souls College, University of Oxford*.

https://www.asc.ox.ac.uk/person/professor-sir-simon-donaldson - Simon Donaldson,
*Simons Center for Geometry and Physics*.

https://scgp.stonybrook.edu/people/faculty/bios/simon-donaldson - Simon Donaldson, The Shaw Prize 2009,
*The Shaw Foundation*(2009).

https://www.shawprize.org/laureates/2009-mathematical-sciences/ - Simon Donaldson (Royal Society Research Professor),
*Department of Mathematics, Imperial College London*.

https://www.ma.imperial.ac.uk/~skdona/ - Simon Donaldson, Five Winners Receive Inaugural Breakthrough Prize In Mathematics,
*The Breakthrough Prize*.

https://breakthroughprize.org/News/18 - Simon Donaldson,
*The Royal Society*.

https://royalsociety.org/people/simon-donaldson-11349/ - Simon Donaldson,
*The Academy of Europe*.

https://www.ae-info.org/ae/Member/Donaldson_Simon - Simon Donaldson: 2015 Breakthrough Prize in Mathematics,
*Mathematics Breakthrough Prize*(2015).

https://breakthroughprize.org/Laureates/3/L55 - Simon Donaldson,
*Alchetron*(4 July 2022).

https://alchetron.com/Simon-Donaldson#Biography - Simon K Donaldson: Wolf Prize Laureate in Mathematics 2020,
*Wolf Foundation*(13 January 2020).

https://wolffund.org.il/2020/01/13/simon-donaldson/ - Simon K Donaldson,
*Institute for Advanced Study*.

https://www.ias.edu/scholars/simon-k-donaldson - Simon K Donaldson,
*National Academy of Sciences*.

https://www.nasonline.org/member-directory/members/3009913.html - Simon Kirwan Donaldson,
*encyclopedia.com*.

https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/simon-kirwan-donaldson - Simon Kirwan Donaldson,
*Prabook*.

https://prabook.com/web/simon.donaldson/3745317 - Sir Simon Donaldson,
*Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics*.

https://sites.duke.edu/scshgap/simon-donaldson/ - Sir Simon Donaldson Awarded Wolf Prize,
*London Mathematical Society*(2020).

https://www.lms.ac.uk/news-entry/15012020-1129/sir-simon-donaldson-awarded-wolf-prize - R J Stern, Review: The geometry of four-manifolds, by S K Donaldson and P B Kronheimer,
*Mathematical Reviews*MR1079726**(92a:57036)**. - R Stern and G Tian, Donaldson and Yau receive Crafoord prize,
*Notices of the American Mathematical Society***41**(7) (1994), 794-796. - The Royal Swedish Academy of Sciences awards The 1994 Crafoord Prize to differential geometry,
*Royal Swedish Academy of Sciences*(12 January 1994).

https://www.kva.se/en/news/vetenskapsakademien-utdelar-1994-ars-crafoordpris-inom-differentialgeometri-2/ - The work of Prof Simon Donaldson,
*Universidad Complutense Madrid*.

https://matematicas.ucm.es/work-simon-donaldson - The Universe Speaks in Numbers: Simon Donaldson interviewed by Graham Farmelo,
*Apple Podcasts*24 October 2019).

https://podcasts.apple.com/gb/podcast/the-universe-speaks-in-numbers-simon-donaldson/id1479389232?i=1000454765326 - A Zvonkin, Review: Riemann surfaces, by Simon Donaldson,
*Mathematical Reviews*MR2856237. - A Jackson, Donaldson and Narasimhan Receive 2006 King Faisal Prize,
*Notices of the American Mathematical Society***41**(7) (2006), 347. - M Jones, Imperial mathematician wins one of the world's richest scientific prizes,
*Imperial College London*(24 June 2014). - J Hassan and N Al-Shehri, King Faisal Prize Winners Announced,
*Arab News*(28 December 2005). - Nemmers Awards in Economics, Mathematics Announced,
*Northwestern University*(22 April 2008).

https://www.northwestern.edu/newscenter/stories/2008/04/nemmersprizes.html - Contribution of Simon K Donaldson & Clifford H Taubes,
*The Shaw Foundation*(7 October 2009).

https://www.shawprize.org/laureates/2009-mathematical-sciences/?type=Contribution

### Additional Resources (show)

Other pages about Simon Donaldson:

Other websites about Simon Donaldson:

### Honours (show)

Honours awarded to Simon Donaldson

- Lecturer at the EMS
- Bowen Lecturer 1997-98
- LMS Whitehead Prize 1984
- BMC morning speaker 1986
- Speaker at International Congress 1986
- Fields Medal 1986
- Fellow of the Royal Society 1986
- Royal Society Royal Medal winner 1992
- The Crafoord Prize 1994
- LMS Pólya Prize 1999
- BMC plenary speaker 2000
- Shaw Prize 2009
- The Breakthrough Prize in Mathematics 2015
- International Congress Speaker 2018
- Oswald Veblen Prize of the AMS 2019
- Wolf Prize 2020

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update December 2023

Last Update December 2023